Critical-point criterion for equal conformal connections (${\tilde{\nabla }}_p={\nabla }_p⟺(d\sigma {)}_p=0$)

On a positive-dimensional Riemannian manifold, equality means equality of the two covariant derivatives at $p$ for every pair of vector fields. A critical point of $\sigma$ annihilates the conformal correction. Conversely, writing that correction as $A$, the identity $g(A(v,v),v)=g(v,v)d\sigma (v)$ for all tangent vectors forces $d\sigma =0$ if $A=0$. A smooth function on a nonempty compact boundaryless manifold has a critical point, so conformally related connections coincide somewhere. Compact manifolds with boundary need not have one.